Images and properties of cotangent function

In a right-angled triangle, the ratio of the adjacent right-angled side to the opposite right-angled side of an acute angle is called the cotangent of the acute angle [1]. Cotangent and tangent are reciprocal, which is expressed by "cot+ angle". The image of cotangent function consists of some isolated branches (as shown in the figure). Cotangent function is an unbounded function, which can take a practical value. It is also a odd function sum periodic function, and its minimum positive period is π[2].

Chinese name

cotangent

Foreign name

cotangent

Simplified Chinese characters

cottage

definition

The ratio of adjacent right-angled sides to opposite sides of an acute angle.

discipline classification

Mathematical science

quick

navigate by water/air

Ethnicity

Images and attributes

Operational relationship

Cotangent sequence

definition

The abscissa of any point on the terminal edge of any angle except the vertex is divided by the non-zero ordinate of the point, and the vertex of the angle coincides with the origin of the plane rectangular coordinate system, while the starting edge of the angle coincides with the positive direction of the X axis. To put it simply: the ratio of the adjacent side to the opposite side of any acute angle of a right triangle is called the cotangent of the acute angle.

Graph 1 cotangent graph

Cotangent is represented by "cot+ angle", for example, the cotangent of 30 is represented by COT 30 and the cotangent of angle α is represented by cota. In the old days, cotangent was expressed by ctg A, which is the same as COT A. Assuming that the opposite side of ∠A is A and the adjacent side is B, cot A= b/a (that is, the adjacent side is better than the edge) [1].

Ethnicity

The Syrian astronomer and mathematician Al Albatani (850-929) made a cotangent table with an interval of 1 degree from 0 to 90 degrees around 920.

/kloc-In the middle of the 4th century, Arubo (1393- 1449), a descendant of Genghis Khan, organized large-scale astronomical observation and calculation of mathematical tables. His sine table is accurate to 9 decimal places, and he also made 1 "for the interval between 30 degrees and 45 degrees and the interval between 45 degrees and 90 degrees.

The British mathematician and Archbishop of Canterbury, Bravadin (1290- 1349) first introduced tangent and cotangent into his trigonometric calculation [3].

Images and attributes

The function image of cotangent function is shown in Figure 2, and its main properties are as follows:

Fig. 2 image of cotangent function

(1) domain: the domain of cotangent function is;

(2) Range: the range of cotangent function is real number set R, and there is no maximum and minimum;

(3) periodicity: the cotangent function is a periodic function with a period of;

(4) parity: the cotangent function is odd function, and its image is symmetrical about the origin;

(5) Monotonicity: the cotangent function is a decreasing function in every open interval [4].

Operational relationship

Harmony and harmonious relationship

Product relationship

Relationship of quotient

Then from Taylor series.

Sum angle formula

Cotangent sequence

Cotangent sequence is a typical example of butterfly effect. Each of the following three series is the cotangent of the previous series, that is, the initial values are 1, 1.0000 1, 1.000 1 respectively, but starting from 10, the three series This is a chaotic sequence. After enough input, the obtained numbers can be considered as random and chaotic.

first

second

The third/third in ten days' work

1.0000 1

1.000 1

0.6420926 16

0.642078493

0.64 195 1397

Spread out completely

reference data

[1] Department of Applied Mathematics, Tongji University. Advanced mathematics. Higher Education Press, 2007

[2] High edition. Dictionary of detailed explanation of mathematical terms. Shaanxi Science and Technology Press, 199 1: 72 1.